Bifurcations for a FitzHugh-Nagumo Model with Time Delays
In this paper, a FitzHugh-Nagumo model with time delays is investigated. The linear stability of the equilibrium and the existence of Hopf bifurcation with delay τ is investigated. By applying Nyquist criterion, the length of delay is estimated for which stability continues to hold. Numerical simulations for justifying the theoretical results are illustrated. Finally, main conclusions are given.
<p>[1] G.S. Medvedev, N. Kopell, Synchronization and transient dynamics in
the chains of electrically coupled FitzHugh-Nagumo oscillators. SIAM J.
Appl. Math. 61 (2001) 1762-1801.
[2] H. Freedman, V. S. H. Rao, The trade-off between mutual interference
and time lags in predator-prey systems. Bull. Math. Biol. 45(6)(1983)
991-1004.
[3] J. Hale, Theory of Functional Differential Equation. Springer-Verlag,
1977.
[4] T. Kostova, R. Ravindran, M. Schonbek, FitzHugh-Nagumo revisted:Type
of bifurcations, periodical forcing and stability regions by Lyapunov
functional. Int. J. Bifur. Chaos Appl. Sci. Engrg. 14 (2004) 913-925.
[5] Y. Kuang, Delay Differential Equations With Applications in Population
Dynamics. Academic Press, INC, 1993.
[6] M. Ringkvist, Y. Zhou, On the dynamical behaviour of FitzHugh-Nagumo
systems: Revisited. Nonlinear Anal. 71 (2009) 2667-2687.
[7] S. G. Ruan, J. J. Wei, On the zero of some transcendential functions with
applications to stability of delay differential equations with two delays.
Dyn. Contin. Discrete Impuls. Syst. Ser. A 10(1) (2003) 863-874.</p>
<p>[1] G.S. Medvedev, N. Kopell, Synchronization and transient dynamics in
the chains of electrically coupled FitzHugh-Nagumo oscillators. SIAM J.
Appl. Math. 61 (2001) 1762-1801.
[2] H. Freedman, V. S. H. Rao, The trade-off between mutual interference
and time lags in predator-prey systems. Bull. Math. Biol. 45(6)(1983)
991-1004.
[3] J. Hale, Theory of Functional Differential Equation. Springer-Verlag,
1977.
[4] T. Kostova, R. Ravindran, M. Schonbek, FitzHugh-Nagumo revisted:Type
of bifurcations, periodical forcing and stability regions by Lyapunov
functional. Int. J. Bifur. Chaos Appl. Sci. Engrg. 14 (2004) 913-925.
[5] Y. Kuang, Delay Differential Equations With Applications in Population
Dynamics. Academic Press, INC, 1993.
[6] M. Ringkvist, Y. Zhou, On the dynamical behaviour of FitzHugh-Nagumo
systems: Revisited. Nonlinear Anal. 71 (2009) 2667-2687.
[7] S. G. Ruan, J. J. Wei, On the zero of some transcendential functions with
applications to stability of delay differential equations with two delays.
Dyn. Contin. Discrete Impuls. Syst. Ser. A 10(1) (2003) 863-874.</p>
@article{"International Journal of Engineering, Mathematical and Physical Sciences:65274", author = "Changjin Xu and Peiluan Li", title = "Bifurcations for a FitzHugh-Nagumo Model with Time Delays", abstract = "In this paper, a FitzHugh-Nagumo model with time delays is investigated. The linear stability of the equilibrium and the existence of Hopf bifurcation with delay τ is investigated. By applying Nyquist criterion, the length of delay is estimated for which stability continues to hold. Numerical simulations for justifying the theoretical results are illustrated. Finally, main conclusions are given.
", keywords = "FitzHugh-Nagumo model, Time delay, Stability, Hopf
bifurcation.", volume = "7", number = "1", pages = "153-5", }